TY - JOUR

T1 - Constructing a polynomial whose nodal set is the three-twist knot 5_2

AU - Dennis, Mark

AU - Bode, Benjamin

N1 - arXiv: 1612.06801

PY - 2017/6/6

Y1 - 2017/6/6

N2 - We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot 52. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain complex polynomial which depends on an additional parameter. For sufficiently small values of this parameter, the nodal lines form the three-twist knot. Further mathematical properties of this map are explored, including the relationship of the phase critical points with the Morse–Novikov number, which is nonzero as this knot is not fibred. We also find analogous functions for other simple knots and links. The particular function we find, and the general procedure, should be useful for designing knotted fields of particular knot types in various physical systems.

AB - We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot 52. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain complex polynomial which depends on an additional parameter. For sufficiently small values of this parameter, the nodal lines form the three-twist knot. Further mathematical properties of this map are explored, including the relationship of the phase critical points with the Morse–Novikov number, which is nonzero as this knot is not fibred. We also find analogous functions for other simple knots and links. The particular function we find, and the general procedure, should be useful for designing knotted fields of particular knot types in various physical systems.

UR - https://research-information.bristol.ac.uk/en/publications/constructing-a-polynomial-whose-nodal-set-is-the-threetwist-knot-52(b0ccf467-9c1e-4196-a2ea-b897740ddada).html

U2 - 10.1088/1751-8121/aa6cbe

DO - 10.1088/1751-8121/aa6cbe

M3 - Article

SN - 1751-8113

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 26

M1 - 265204

ER -